This talk concerns the dynamics of an SIS reaction–diffusion model with standard incidence in a spatially heterogeneous environment  proposed in L.J.S. Allen, B.M. Bolker, Y. Lou and A.L. Nevai, DCDS, 2008. We focus on the stability of endemic equilibria and the existence of time-periodic solutions. We first present a local stability criterion and its applications to several diffusion regimes. We then discuss recent progress on periodic dynamics. On a one-dimensional interval, we construct smooth, time-independent transmission and recovery rates for which strictly positive, nonstationary time-periodic solutions exist. Finally, we compare these results with the corresponding SIS patch model.